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Enterprise Modelling and Information Systems ArchitecturesVol. 0, No. 0 (month 0000).Representing Imprecise Time Intervals in OWL 2 1

Representing Imprecise Time Intervals in OWL 2

Elisabeth Métais*,a, Fatma Ghorbelb, Fayçal Hamdia, Nebrasse Ellouzeb, NouraHerradia, Assia Soukanec

a CEDRIC Laboratory, Conservatoire National des Arts et Métiers, Paris, Franceb MIRACL Laboratory, University of Sfax, Sfax, Tunisiac Ecole Centrale d’Electronique, Paris, France

Abstract. Representing and reasoning on imprecise temporal information is a common requirement inthe field of Semantic Web. Many works exist to represent and reason on precise temporal information inOWL; however, to the best of our knowledge, none of these works is devoted to imprecise temporal timeintervals. To address this problem, we propose two approaches: a crisp-based approach and a fuzzy-basedapproach. (1) The first approach uses only crisp standards and tools and is modeled in OWL 2. We extendthe 4D-fluents model, with new crisp components, to represent imprecise time intervals and qualitative crispinterval relations. Then, we extend the Allen’s interval algebra to compare imprecise time intervals in acrisp way and inferences are done via a set of SWRL rules. (2) The second approach is based on fuzzy setstheory and fuzzy tools and is modeled in Fuzzy-OWL 2. The 4D-fluents approach is extended, with newfuzzy components, in order to represent imprecise time intervals and qualitative fuzzy interval relations.The Allen’s interval algebra is extended in order to compare imprecise time intervals in a fuzzy gradualpersonalized way. Inferences are done via a set of Mamdani IF-THEN rules.

Keywords. Imprecise time interval • Linked data • OWL • Allen’s interval algebra • 4D-fluents

1 Introduction

In the Semantic Web field, representing and reas-oning on imprecise temporal information is a com-mon requirement. Indeed, temporal informationgiven by users is often imprecise. For instance, ifthey give the information “Alexandre was marriedto Nicole by 1981 to late 90” two measures ofimprecision are involved. On the one hand, theinformation “by 1981” is imprecise in the sensethat it could mean approximately from 1980 to1982; on the other hand, the information “late 90”is imprecise in the sense that it could mean, withan increasingly possibility, from 1995 to 2000.When an event is characterized by a gradual be-ginning and/or ending, it is usual to represent thecorresponding time span as an imprecise timeinterval.

* Corresponding author.E-mail. [email protected]

In OWL, many works have been proposed torepresent and reason on precise temporal informa-tion; however, to the best of our knowledge, thereis no work devoted to represent and reason onimprecise temporal time intervals. In this paper,we answer this problem with two approaches, oneusing a crisp environment, the other one using afuzzy environment.

The first approach involves only crisp standardsand tools. To represent imprecise time intervals inOWL 2, we extend the so called 4D-fluents model(Welty and Fikes 2006) which is a formalismto model crisp quantitative temporal informationand the evolution of temporal concepts in OWL.This model is extended in two ways: (1) It isenhanced with new crisp components for modelingimprecise time intervals. (2) It is enhanced withqualitative temporal expressions representing crisprelations between imprecise temporal intervals. To

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Enterprise Modelling and Information Systems ArchitecturesVol. 0, No. 0 (month 0000).

2 Elisabeth Métais, Fatma Ghorbel, Fayçal Hamdi, Nebrasse Ellouze, Noura Herradi, Assia Soukane

reason on imprecise time intervals, we extend theAllen’s interval algebra (Allen 1983) which is themost used and known formalisms for reasoningabout crisp time intervals. We generalize Allen’srelationships to handle imprecise time intervalswith a crisp view. The resulting crisp temporalinterval relations are inferred from the introducedimprecise time intervals using a set of SWRL rules(Horrocks et al. 2004), in OWL 2.

The second approach is based on fuzzy sets the-ory and fuzzy tools. It is based on Fuzzy-OWL 2(Bobillo and Straccia 2011) which is an extensionof OWL 2 that deals with fuzzy information. Torepresent imprecise time intervals in Fuzzy-OWL2, we extend the 4D-fluents model in two ways:(1) It is enhanced with new fuzzy components tobe able to model imprecise time intervals. (2) Itis enhanced with qualitative temporal expressionsrepresenting fuzzy relations between imprecisetemporal intervals. To reason on imprecise timeintervals, we extend Allen’s work to compareimprecise time intervals in a fuzzy gradual per-sonalized way. Our Allen’s extension introducesgradual fuzzy interval relations e.g., “long before”.It is personalized in the sense that it is not limited toa given number of interval relations. It is possibleto determinate the level of precision that should bein a given context. For instance, the classic Allenrelation “before” may be generalized in N intervalrelations, where “before(1)” means “just before”and gradually the time gap between the two impre-cise intervals increases until “before(N)” whichmeans “long before”. The resulting fuzzy intervalrelations are inferred from the introduced impre-cise time intervals using the FuzzyDL reasoner(Bobillo and Straccia 2008), via a set of MamdaniIF-THEN rules, in Fuzzy-OWL 2.

The current paper is organized as follows: Sec-tion 2 is devoted to present some preliminaryconcepts and related work in the field of temporalinformation representation in OWL and reasoningon time intervals. In Section 3, we introduce ourcrisp-based approach for representing and reas-oning on imprecise time intervals. In Section 4,we introduce our fuzzy-based approach for repres-enting and reasoning on imprecise time intervals.

Section 5 draws conclusions and future researchdirections.

2 Preliminaries and Related WorkIn this section, we introduce some preliminaryconcepts and related work in the field of temporalinformation representation in OWL and reasoningon time intervals.

2.1 Representing Temporal Informationin OWL

Five main approaches are proposed to representtime information in OWL: Temporal DescriptionLogics (Artale and Franconi 2000), Versioning(Klein and Fensel 2001), N-ary relations (Noyand Rector 2006) and 4D-fluents (Welty and Fikes2006). All these approaches represent only crisptemporal information in OWL. Temporal Descrip-tion Logics extend the standard description logicswith additional temporal constructs e.g., “some-time in the future”. N-ary relations approachrepresents an N-ary relation using an additionalobject. The N-ary relation is represented as twoproperties each related with the new object. Thetwo objects are related to each other with an N-ary relation. Reification is “a general purposetechnique for representing N-ary relations using alanguage such as OWL that permits only binaryrelations” (Batsakis and Petrakis 2011). Version-ing approach is described as “the ability to handlechanges in ontologies by creating and managingdifferent variants of it” (Klein and Fensel 2001).When an ontology is modified, a new versionis created to represent the temporal evolution ofthe ontology. 4D-fluents approach represents tem-poral information and the evolution of the last onesin OWL. Concepts varying in time are representedas 4-dimensional objects with the 4th dimensionbeing the temporal dimension.

Based on the present related work, we choosethe 4D-fluents approach. Indeed, compared torelated work, it minimizes the problem of dataredundancy as the changes occur only on the tem-poral parts and keeping therefore the static partunchanged. It also maintains full OWL expressive-ness and reasoning support (Batsakis and Petrakis


2011). We extend this approach in two ways. (1)It is extended with crisp components to representimprecise time intervals and crisp interval rela-tions in OWL 2 (Section 3). (2) It is extended withfuzzy components to represent imprecise time in-tervals and fuzzy interval relations in Fuzzy-OWL2 (Section 4).

2.2 Allen’s Interval Algebra(Allen 1983) has proposed 13 mutually exclusiveprimitive relations that may hold between two pre-cise time intervals. Their semantics is illustratedin Table 1. Let I = [I−, I+] and J = [J−, J+]two time intervals; where I− (respectively J−)is the beginning time-step of the event and I+

(respectively J−) is the ending.A number of works fuzzify Allen’s temporal

interval relations. We classify these works into(1) works focusing on fuzzifying Allen’s intervalalgebra to compare precise time intervals and(2) works focusing on fuzzifying Allen’s intervalalgebra to compare imprecise time intervals.

Three approaches have been proposed to fuzzifyAllen’s interval algebra in order to compare precisetime intervals: (Guesgen et al. 1994), (Duboisand Prade 1989) and (Badaloni and Giacomin2006). (Guesgen et al. 1994) propose fuzzy Allenrelations viewed as fuzzy sets of ordinary Allenrelationship taking into account a neighborhoodstructure, a notion originally introduced in (Freksa1992). (Dubois and Prade 1989) represent a timeinterval as a pair of possibility distributions thatdefine the possible values of the endpoints ofthe crisp interval. Using possibility theory, thepossibility and necessity of each of the intervalrelations can then be calculated. This approachalso allows modeling imprecise relations such as“long before”. (Badaloni and Giacomin 2006)propose a fuzzy extension of Allen’s work, calledIAfuz where degrees of preference are associatedto each relation between two precise time intervals.

Four approaches have been proposed to fuzzifyAllen’s interval algebra to compare imprecise timeintervals: (Nagypál and Motik 2003), (Ohlbach2004), Schockaert08 and (Gammoudi et al. 2017).(Nagypál and Motik 2003) propose a temporal

model based on fuzzy sets to extend Allen rela-tions with imprecise time intervals. The authorsintroduce a set of auxiliary operators on intervalsand define fuzzy counterparts of these operators.The compositions of these relations are not studiedby the authors. (Ohlbach 2004) propose an ap-proach to handle some gradual temporal relationsas “more or less finishes”. However, this work can-not take into account gradual temporal relationssuch as “long before”. Furthermore, many of thesymmetry, reflexivity, and transitivity propertiesof the original temporal interval relations are lostin this approach; thus it is not suitable for temporalreasoning. (Schockaert and Cock 2008) proposea generalization of Allen’s relations with preciseand imprecise time intervals. This approach al-lows handling classical temporal relations, as wellas other imprecise relations. Interval relations aredefined according to two fuzzy operators compar-ing two time instants: “long before” and “occursbefore or at approximately the same time”. (Gam-moudi et al. 2017) generalize the definitions ofthe 13 Allen’s classic interval relations to makethem applicable to fuzzy intervals in two ways(conjunctive and disjunctive). Gradual temporalinterval relations are not taken into account.

3 A Crisp-Based Approach forRepresenting and Reasoning onImprecise Time Intervals

In this section, we propose a crisp-based approachto represent and reason on imprecise time intervals.This solution is entirely based on crisp standardsand tools. We extend the 4D-fluents model torepresent imprecise time intervals and their crisprelationships in OWL 2. To reason on imprecisetime intervals, we extend the Allen’s interval al-gebra in a crisp way. In OWL 2, inferences aredone via a set of SWRL rules.

3.1 Representing Imprecise TimeIntervals and Crisp QualitativeInterval Relations in OWL 2

In the crisp-based solution, we now represent eachimprecise interval bound of the time interval as



Table 1: Allen’s temporal interval relations (I: , J: ).

Relation Inverse Relations between interval bounds Illustration

Be f ore(I, J) A f ter (I, J) I+ < J−

Meets(I, J) MetBy(I, J) I+ = J−

Overlaps(I, J) OverlappedBy(I, J) (I− < J−) ∧ (I+ > J−) ∧ (I+ < J+)

Starts(I, J) StartedBy(I, J) (I− = J−) ∧ (I+ < J+)

During(I, J) Contains(I, J) (I− > J−) ∧ (I+ < J+)

Ends(I, J) EndedBy(I, J) (I− > J−) ∧ (I+ = J+)

Equal (I, J) Equal (I, J) (I− = J−) ∧ (I+ = J+)

a disjunctive ascending set. Let I = [I−, I+]be an imprecise time interval; where I− =I−(1) . . . I−(N ) and I+ = I+(1) . . . I+(N ). For in-stance, if we have the information “Alexandre wasstarted his PhD study in 1975 and he was graduatedaround 1980” the imprecise time interval repres-enting this period is [1975, {1978 . . . 1982}]. Thismeans that his PhD studies end in 1978 or 1979or 1980 or 1981 or 1982. The classic 4D-fluentsmodel introduces two crisp classes “TimeSlice”and “TimeInterval” and four crisp properties “ts-TimeSliceOf", “tsTimeInterval", “hasBegining”and “hasEnd”. The class “TimeSlice” is the do-main class for entities representing temporal parts(i.e., “time slices"). The property “tsTimeSliceOf”connects an instance of class “TimeSlice” withan entity. The property “tsTimeInterval” con-nects an instance of class “TimeSlice” with aninstance of class “TimeInterval”. The instanceof class “TimeInterval” is related with two tem-poral instants that specify its starting and endingpoints using, respectively, the “hasBegining” and“hasEnd” properties. Figure 1 illustrates the useof the 4D-fluents model to represent the followingexample: “Alexandre was started his PhD studyin 1975 and he was graduated in 1978”.

We extend the original 4D-fluents model in thefollowing way. We add four crisp datatype prop-erties “HasBeginningFrom", “HasBeginningTo",“HasEndFrom", and “HasEndTo” to the class

“TimeInterval”. Let I = [I−, I+] be an impre-cise time interval; where I− = I−(1) . . . I−(N )

and I+ = I+(1) . . . I+(N ). “HasBeginningFrom”has the range I−(1). “HasBeginningTo” has therange I−(N ). “HasEndFrom” has the range I+(1).“HasEndTo” has the range I+(N ) . The 4D-fluentsmodel is also enhanced with crisp qualitative tem-poral interval relations that may hold betweenimprecise time intervals. This is implemented byintroducing temporal relationships, called “Rela-tionIntervals", as a crisp object property betweentwo instances of the class “TimeInterval”. Fig-ure 2 represents the extended 4D-fluents model inOWL 2.

We can see in Figure 3 an instantiation of theextended 4D-fluents model in OWL 2. On thisexample, we consider the following information:“Alexandre was married to Nicole just after he wasgraduated with a PhD. Alexandre was graduatedwith a PhD in 1980. Their marriage lasts 15 years.Alexandre was remarried to Béatrice since about10 years and they were divorced in 2016”. Let I =[I−, I+] and J = [J−, J+] be two imprecise timeintervals representing, respectively, the duration ofthe marriage of Alexandre with Nicole and the onewith Béatrice. Assume that I− = {1980 . . . 1983},I+ = {1995 . . . 1998}, J− = {2006 . . . 2008} andJ+ = 2016.


Figure 1: An instantiation of the classic the 4D-fluents model.

Figure 2: The extended 4D-fluents model in OWL 2.

3.2 A Crisp-Based Reasoning onImprecise Time Intervals in OWL 2

We have redefined 13 Allen’s interval, in a crispway, to compare imprecise time intervals i.e.,the resulting interval relations upon imprecisetime intervals are crisp. Let I = [I−, I+] andJ = [J−, J+] two imprecise time intervals; whereI− = I−(1) . . . I−(N ), I+ = I+(1) . . . I+(N ), J− =J−(1) . . . J−(N ) and J+ = J+(1) . . . J+(N ). Forinstance, the crisp interval relation “before (I, J)”is redefined as: ∀I+(i) ∈ I+,∀J−( j ) ∈ J− : I+(i) <

J−( j ) This means that the most recent time instantof I+(I+(N )) ought to Precede the oldest timeinstant of J−(J−(1)): I+(N ) < J−(1) In the similar

way, we define the other temporal interval relations.In Table 2, we define 13 crisp temporal intervalrelations upon the two imprecise time intervals Iand J.

In order to apply our crisp extension of Allen’swork in OWL 2, we propose a set of SWRL rulesthat infer the temporal interval relations from theintroduced imprecise time intervals which arerepresented using the extended 4D-fluents modelin OWL2. For each temporal interval relation, weassociate a SWRL rule. Reasoners that supportDL-safe rules (i.e., rules that apply only on namedindividuals in the knowledge base) such as Pellet(Sirin et al. 2007) can support our approach. For



Figure 3: An instantiation of the extended 4D-fluents model in OWL 2.

instance, the SWRL rule to infer the “Meet (I, J)”relation is the following:

TimeInterval (I) ∧ TimeInterval (J) ∧HasEndFrom(I, a) ∧

HasBeginningFrom(J, b) ∧ Equals(a, b) ∧HasEndTo(I, c) ∧ HasBeginningTo(J, d) ∧

Equals(c, d) → Meet(I, J)

4 A Fuzzy-Based Approach forRepresenting and Reasoning onImprecise Time Intervals

In this section, we propose a fuzzy-based approachto represent and reason on imprecise time intervals.This approach is based on a fuzzy environment.We extend the 4D-fluents model to represent im-precise time intervals and their relationships inFuzzy-OWL 2. To reason on imprecise time in-tervals, we extend the Allen’s interval algebra ina fuzzy gradual personalized way. We infer theresulting fuzzy interval relations in Fuzzy-OWL 2using a set of Mamdani IF-THEN rules.

4.1 Representing Imprecise TimeIntervals and Fuzzy QualitativeInterval Relations in Fuzzy-OWL 2

In the fuzzy-based solution, we now represent theimprecise beginning interval bound as a fuzzy setwhich has the L-function MF and the ending inter-val bound as a fuzzy set which has the R-functionmembership function (MF). Let I = [I−, I+] be animprecise time interval. We represent the binging

bound I− as a fuzzy set which has the L-functionMF (A = I−(1) and B = I−(N )). We representthe ending bound I+ as a fuzzy set which has theR-function MF (A = I+(1) and B = I+(N )). Forinstance, if we have the information “Alexandrewas starting his PhD study in 1973 and was gradu-ated in late 80", the beginning bound is crisp. Theending bound is imprecise and it is representedby L-function MF (A = 1976 and B = 1980). Forthe rest of the paper, we use the MFs shown inFigure 4 [Zadeh, 1975].

We extend the original 4D-fluents model torepresent imprecise time intervals in the follow-ing way. We add two fuzzy datatype properties“FuzzyHasBegining” and “FuzzyHasEnd” to theclass “TimeInterval”. “FuzzyHasBegining” hasthe L-function MF (A = I−(1) and B = I−(N )).“FuzzyHasEnd” has the R-function MF (A = I+(1)

and B = I+(N )). The 4D-fluents approach isalso enhanced with qualitative temporal relationsthat may hold between imprecise time intervals.We introduce the “FuzzyRelationIntervals", asa fuzzy object property between two instancesof the class “TimeInterval”. “FuzzyRelation-Intervals” represent fuzzy qualitative temporalrelations. “FuzzyRelationIntervals” has the L-function MF (A = 0 and B = 1). Figure 5 repres-ents our extended 4D-fluents model in Fuzzy-OWL2.

We can see in Figure 6 an instantiation of the ex-tended 4D-fluents model in Fuzzy-OWL 2. On this


Table 2: Crisp temporal interval relations upon imprecise time intervals.

Relation Inverse Interpretation Relations between interval bounds

Be f ore(I, J ) Af ter (I, J ) ∀I+(i) ∈ I+, ∀J−( j ) ∈ J− :(I+(i) < J−( j ) )

I+(N ) < J−(1)

Meets(I, J ) MetBy (I, J ) ∀I+(i) ∈ I+, ∀J−( j ) ∈ J− :(I+(i) = J−( j ) )

(I+(1) = J−(1) ) ∧ (I+(N ) = J−(N ) )

Overlaps(I, J ) OverlappedBy (I, J ) ∀I−(i) ∈ I−, ∀I+(i) ∈

I+, ∀J−( j ) ∈ J−, ∀J+( j ) ∈ J+ :(I−(i) < J−( j ) ) ∧ (J−( j ) <I+(i) ) ∧ (I+(i) < J+( j ) )

(I−(N ) < J−(1) ) ∧ (J−(N ) <I+(1) ) ∧ (I+(N ) < J+(1) )

Star t s(I, J ) Star tedBy (I, J ) ∀I−(i) ∈ I−, ∀I+(i) ∈

I+, ∀J−( j ) ∈ J−, ∀J+( j ) ∈ J+ :(I−(i) = J−( j ) ) ∧ (I+(i) < J+( j ) )

(I−(1) = J−(1) ) ∧ (I−(N ) =

J−(N ) ) ∧ (I+(N ) < J+(1) )

During(I, J ) Contains(I, J ) ∀I−(i) ∈ I−, ∀I+(i) ∈

I+, ∀J−( j ) ∈ J−, ∀J+(i) ∈ J+ :(J−( j ) < I−(i) ) ∧ (I+(i) < J+(i) )

(J−(N ) < I−(1) ) ∧ (I+(N ) < J+(1) )

Ends(I, J ) EndedBy (I, J ) ∀I−(i) ∈ I−, ∀I+(i) ∈

I+, ∀J−( j ) ∈ J−, ∀J+( j ) ∈ J+ :(I−(i) < J−( j ) ) ∧ (I+(i) = J+( j ) )

(J−(N ) < I−(1) ) ∧ (I+(1) =

J+(1) ) ∧ (I+(N ) = J+(N ) )

Equal (I, J ) Equal (I, J ) ∀I−(i) ∈ I−, ∀I+(i) ∈

I+, ∀J−( j ) ∈ J−, ∀J+( j ) ∈ J+ :(I−(i) = J−( j ) ) ∧ (I+(i) = J+( j ) )

(I−(1) = J−(1) ) ∧ (I−(N ) = J−(N ) ) ∧(I+(1) = J+(1) ) ∧ (I+(N ) = J+(N ) )

example, we consider the following information:“Alexandre was married to Nicole just after he wasgraduated with a PhD. Alexandre was graduatedwith a PhD in 1980. Their marriage lasts 15 years.Alexandre was remarried to Béatrice since about10 years and they were divorced in 2016”. LetI = [I−, I+] and J = [J−, J+] be two imprecisetime intervals representing, respectively, the dur-ation of the marriage of Alexandre with Nicoleand the one with Béatrice. I− is represented withthe fuzzy datatype property “FuzzyHasBegining”which has the L-function MF (A = 1980 andB = 1983). I+ is represented with the fuzzydatatype property “FuzzyHasEnd” which has theR-function MF (A = 1995 and B = 1998). J−

is represented with the fuzzy datatype property“FuzzyHasBegining” which has the L-functionMF (A = 2005 and B = 2007). J+ is representedwith the crisp datatype property “HasEnd” whichhas the value “2016”.

4.2 A Fuzzy-Based Reasoning onImprecise Time Intervals in FuzzyOWL 2

We propose a set of fuzzy gradual personalizedcomparators that may hold between two timeinstants. Based on these operators, we present ourfuzzy gradual personalized extension of Allen’swork. Then, we infer, in Fuzzy OWL 2, theresulting temporal interval relations via a set ofMamdani IF-THEN rules using the fuzzy reasonerFuzzyDL. We generalize the crisp time instantscomparators “Follow", “Precede” and “Same",introduced in (Vilain and Kautz 1986). Let α andβ two parameters allowing the definition of themembership function of the following comparators(∈]0,+∞[); N is the number of slices; T1 and T2are two time instants; we define the followingcomparators (illustrated in Figure 7):

• {Follow(α,β)(1) (T1,T2) . . . Follow(α,β)

(N ) (T1,T2)}are a generalization of the crisp time in-stants relation “Follows”. Follow(α,β)

(1) (T1,T2)means that T1 is “just after or approxim-ately at the same time” T2 w.r.t. (α, β)and gradually the time gap between T1



Figure 4: R-Function, L-Function and Trapezoidal MFs (Zadeh 1975).

and T2 increases until Follow(α,β)(N ) (T1,T2)

which means that T1 is “long after” T2w.r.t. (α, β). N is set by the expert domain.{Follow(α,β)

(1) (T1,T2) . . . Follow(α,β)(N ) (T1,T2)}

are defined as fuzzy sets. Follow(α,β)(1) (T1,T2)

has R-Function MF which has as parametersA = α and B = (α + β). All comparators{Follow(α,β)

(2) (T1,T2) . . . Follow(α,β)(N−1) (T1,T2)}

have trapezoidal MF which has as parametersA = ((K−1)α) and B = ((K−1)α+ (K−1) β),C = (Kα + (K − 1) β) and D = (Kα + K β);where 2 ≤ K ≤ N − 1. Follow(α,β)

(N ) (T1,T2)has L-Function MF which has as para-meters A = ((N − 1)α + (N − 1) β) andB = ((N − 1)α + (N − 1) β);

• {Precede(α,β)(1) (T1,T2) . . . Precede(α,β)

(N ) (T1,T2)}are a generalization of the crisp time instantsrelation “Precede”. Precede(α,β)

(1) (T1,T2)means that T1 is “just before or approxim-ately at the same time” T2 w.r.t. (α, β)and gradually the time gap between T1and T2 increases until Precede(α,β)

(N ) (T1,T2)which means that T1 is “long before” T2w.r.t. (α, β). N is set by the expert domain.{Precede(α,β)

(1) (T1,T2) . . . Precede(α,β)(N ) (T1,T2)}

are defined as fuzzy sets. Precede(α,β)(i) (T1,T2)

is defined as:

Precede(α,β)(i) (T1,T2) = 1−Follow(α,β)

(i) (T1,T2)

• We define the comparator Same(α,β) whichis a generalization of the crisp time instantsrelation “Same”. Same(α,β) (T1,T2) means thatT1 is “approximately at the same time” T2 w.r.t.(α, β). It is defined as:

Same(α,β) (T1,T2) =Min(Follow(α,β)(1) (T1,T2),

Precede(α,β)(1) (T1,T2))

Then, we extend Allen’s work to compare im-precise time intervals with a fuzzy gradual person-alized view. We provide a way to model gradual,linguistic-like description of temporal interval re-lations. Compared to related work, our work isnot limited to a given number of imprecise rela-tions. It is possible to determinate the level ofprecision that should be in a given context. Forinstance, the classic Allen relation “before” maybe generalized in N imprecise relations, where“Be f ore(α,β)

(1) (I, J)” means that I is “just before”J w.r.t. (α, β) and gradually the time gap betweenI and J increases until “Be f ore(α,β)

(N ) (I, J)” which


Figure 5: The extended 4D-fluents model in Fuzzy-OWL 2.

Figure 6: An instantiation of the extended 4D-fluents model in Fuzzy-OWL 2.

means that I is long before J w.r.t. (α, β). Thedefinition of our fuzzy interval relations is basedon the fuzzy gradual personalized time instantscompactors. Let I = [I−, I+] and J = [J−, J+]two imprecise time intervals; where I− has theL-function MF (A = I−(1) and B = I−(N )); I+ is afuzzy set which has the R-function MF (A = I+(1)

and B = I+(N )); J− is a fuzzy set which has the L-function MF (A = J−(1) and B = J−(N )); J+ is afuzzy set which has the R-function MF (A = J+(1)

and B = J+(N )). For instance, the fuzzy intervalrelation “Be f ore(α,β)

(1) (I, J)” is defined as:

∀I+(i) ∈ I+,∀J−( j ) ∈ J− :

Precede(α,β)(1) (I+(i), J−( j ))

This means that the most recent time instant ofI+(I+(N )) ought to proceed the oldest time instantof J−(J−(1)):

Precede(α,β)(1) (I+(N ), J−(1))

In the similar way, we define the others temporalinterval relations, as shown in Table 3.

Finally, we have implemented our fuzzy gradualpersonalized extension of Allen’s work in Fuzzy-OWL 2. We use the ontology editor PROTEGEversion 4.3 and the fuzzy reasoner FuzzyDL. Wepropose a set of Mamdani IF-THEN rules to in-fer the temporal interval relations from the in-troduced imprecise time intervals which are rep-resented using the extended 4D-fluents model in



Figure 7: Fuzzy gradual personalized time instants comparators. (A) Fuzzy sets of{Follow(α,β)

(1) (T1,T2) . . . Follow(α,β)(N ) (T1,T2)}. (B) Fuzzy sets of {Precede(α,β)

(1) (T1,T2) . . . Precede(α,β)(N ) (T1,T2)}. (C)

Fuzzy set of Same(α,β) (T1,T2).

Table 3: Fuzzy gradual personalized temporal interval relations upon imprecise time intervals.

Relation Inverse Relations between bounds Definition

Be f ore(α,β)(K ) (I, J ) Af ter

(α,β)(K ) (I, J ) ∀I+(i) ∈ I+, ∀J−( j ) ∈ J− :

(I+(i) < J−( j ) )Precede

(α,β)(K ) (I+(N ), J−(1) )

Meets (α,β) (I, J ) MetBy(α,β) (I, J ) ∀I+(i) ∈ I+, ∀J−( j ) ∈ J− :(I+(i) = J−( j ) )

Min(Same (α,β) (I+(1), J−(1) ) ∧Same (α,β) (I+(N ), J−(N ) ))

Overlaps(α,β)(K ) (I, J ) OverlappedBy

(α,β)(K ) (I, J ) ∀I−(i) ∈ I−, ∀I+(i) ∈

I+, ∀J−( j ) ∈ J−, ∀J+( j ) ∈ J+ :(I−(i) < J−( j ) ) ∧ (J−( j ) <I+(i) ) ∧ (I+(i) < J+( j ) )

Min(Precede (α,β)(K ) (I−(N ), J−(1) )∧

Precede(K )(α,β) (J−(N ), I+(1) ) ∧Precede

(α,β)(K ) (I+(N ), J+(1) ))

Star t s(α,β)(K ) (I, J ) Star tedBy

(α,β)(K ) (I, J ) ∀I−(i) ∈ I−, ∀I+(i) ∈

I+, ∀J−( j ) ∈ J−, ∀J+( j ) ∈ J+ :(I−(i) = J−( j ) ) ∧ (I+(i) < J+( j ) )

Min(Same (α,β) (I−(1), J−(1) ) ∧Same (α,β) (I−(N ), J−(N ) ) ∧Precede

(α,β)(K ) (I+(N ), J+(1) ))

During(α,β)(K ) (I, J ) Contains

(α,β)(K ) (I, J ) ∀I−(i) ∈ I−, ∀I+(i) ∈

I+, ∀J−( j ) ∈ J−, ∀J+(i) ∈ J+ :(J−( j ) < I−(i) ) ∧ (I+(i) < J+(i) )

Min(Precede (α,β)(K ) (J−(N ), I−(1) )∧

Precede(α,β)(K ) (I+(N ), J+(1) ))

Ends(α,β)(K ) (I, J ) EndedBy

(α,β)(K ) (I, J ) ∀I−(i) ∈ I−, ∀I+(i) ∈

I+, ∀J−( j ) ∈ J−, ∀J+( j ) ∈ J+ :(I−(i) < J−( j ) ) ∧ (I+(i) = J+( j ) )

Min(Precede (α,β)(K ) (J−(N ), I−(1) )∧

Same (α,β) (I+(1), J+(1) ) ∧Same (α,β) (I+(N ), J+(N ) ))

Equal (α,β) (I, J ) Equal (α,β) (I, J ) ∀I−(i) ∈ I−, ∀I+(i) ∈

I+, ∀J−( j ) ∈ J−, ∀J+( j ) ∈ J+ :(I−(i) = J−( j ) ) ∧ (I+(i) = J+( j ) )

Min(Same (α,β) (I−(1), J−(1) ) ∧Same (α,β) (I−(N ), J−(N ) ) ∧Same (α,β) (I+(1), J+(1) ) ∧Same (α,β) (I+(N ), J+(N ) ))


Fuzzy-OWL2. For each temporal interval rela-tion, we associate a Mamdani IF-THEN rule. Forinstance, the Mamdani IF-THEN rule to infer the“Overlaps(α,β)

(1) (I, J)” relation is the following:

(define-concept Rule0 (g-and (some Precede(1/1)

Fulfilled) (some Precede(1/2) Fulfilled) Fulfilled)(some Precede(1/3) Fulfilled) (some Overlaps(1)

True))) // Fuzzy rule

We define three input fuzzy variables,named “Precede(1/1)", “Precede(1/2)” and“Precede(1/3)", which have the same MF thanthat of “Precede(α,β)

(1) ”. We define one outputvariable “Overlaps(1)” which has the same mem-bership than that of the fuzzy object property“FuzzyRelationIntervals”. “Precede(1/1)", “Pre-cede(1/2)” and “Precede(1/3)” are instantiatedwith, respectively, (I−(N ) − J−(1)), (J−(N ) − I+(1))and (I+(N ) − J+(1)).

5 Conclusion

In this paper, we proposed two approaches torepresent and reason on imprecise time intervalsin OWL: a crisp-based approach and a fuzzy-basedapproach. (1) The crisp-based approach is basedonly on crisp environment. We extended the 4D-fluents model to represent imprecise time intervalsand crisp interval relations in OWL 2. To reason onimprecise time intervals, we extended the Allen’sinterval algebra in a crisp way. Inferences aredone via a set of SWRL rules. (2) The fuzzy-based approach is entirely based only on fuzzyenvironment. We extended the 4D-fluents modelto represent imprecise time intervals and fuzzyinterval relations in Fuzzy-OWL 2. To reason onimprecise time intervals, we extend the Allen’sinterval algebra in a fuzzy gradual personalizedway. We infer the resulting fuzzy interval relationsin Fuzzy-OWL 2 using a set of Mamdani IF-THENrules.

Concerning the choice between these two ap-proaches (the crisp-based one or the fuzzy-basedone), as a fuzzy ontology is an extension of crispontology, researchers may choose any of our two

approaches for introducing imprecise interval man-agement in their knowledge bases whatever theselatter are crisp or fuzzy. However our fuzzy-basedapproach allows more functionalities, in particularit is suitable to represent and reason on gradualinterval relations such as “middle before” or “ap-proximately at the same time”. Hence, in thecase of manipulating a fuzzy knowledge base, weencourage researchers to choose the fuzzy-basedapproach to model and reason on imprecise timeintervals. The main interest of the crisp-basedapproach is that this solution can be implementedwith classical crisp tools and that the program-mers are not obliged to learn technologies relatedto fuzzy ontology. Considering that crisp toolsand models are more mature and better supportscaling, the crisp-based approach is more suitablefor marketed products.

The works presented in this paper have beentested in two projects, having in common to man-age life logging data: (1) In the VIVA project, weaim to design the Captain Memo memory pros-thesis (Herradi et al. 2015; Métais et al. 2012) forAlzheimer Disease patients. Among other func-tionalities, this prosthesis manages a knowledgebase on the patient’s family tree, using an OWLontology. Imprecise inputs are especially numer-ous when given by an Alzheimer Disease patient.Furthermore, dates are often given in reference toother dates or events. Thus we have been usingthe “fuzzy” solution reported in this paper. Oneinteresting point in this solution is to deal with apersonalized slicing of the person’s life in orderto sort the different events. (2) The QUALHISproject aims to allow Middle Ages specialized his-torians to deal with prosopographical data basesstoring Middle age academic’s career histories.Data come from various archives among Europeand data about a same person are very difficult toalign. Representing and ordering imprecise timeinterval is required to redraw the careers acrossthe different European universities who hosted theperson. We preferred the “crisp” solution in orderto favour the integration within the existing crispontologies and tools, and to ease the managementby Historians.



AcknowledgmentsWe are very grateful to Pr. Heinrich Mayr for allthe energy and the enthusiasm he insufflates inthe international research in Computer Scienceand we are highly impressed by his seminal rolein numerous domains of Conceptual Modelling.Furthermore he also never lost his kindness andhumanity in spite of being one of the greatestresearchers in Europe.

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